Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

460 CHAPTER 15 Bending of Open and Closed, Thin-Walled Beams

whichgives

Ixx=

πd^4

64

(15.39)

Clearlyfromsymmetry

Iyy=

πd^4

64

(15.40)

Usingthetheoremofperpendicularaxes,thepolarsecondmomentofarea,Io,isgivenby

Io=Ixx+Iyy=

πd^4

32

(15.41)

15.4.4 Product Second Moment of Area

Theproductsecondmomentofarea,Ixy,ofabeamsectionwithrespecttoxandyaxesisdefinedby

Ixy=

∫

A

xydA (15.42)

Thus,eachelementofareainthecrosssectionismultipliedbytheproductofitscoordinates,andthe
integrationistakenoverthecompletearea.Althoughsecondmomentsofareaarealwayspositive,since
elements of area are multiplied by the square of one of their coordinates, it is possible forIxyto be
negativeifthesectionliespredominantlyinthesecondandfourthquadrantsoftheaxessystem.Such
asituationwouldariseinthecaseoftheZ-sectionofFig.15.30(a)wheretheproductsecondmoment
ofareaofeachflangeisclearlynegative.
Aspecialcaseariseswhenone(orboth)ofthecoordinateaxesisanaxisofsymmetrysothatfor
anyelementofarea,δA,havingtheproductofitscoordinatespositive,thereisanidenticalelementfor
whichtheproductofitscoordinatesisnegative(Fig.15.30(b)).Summation(i.e.,integration)overthe

Fig.15.30

Product second moment of area.